1 4
The Logic of Exclusion: A Comprehensive Analysis of Sudoku Locked Candidates
1. Introduction: Sudoku as a Constraint Satisfaction System
The game of Sudoku, while popularized as a logic puzzle accessible to the layperson, represents a sophisticated exercise in finite domain constraint satisfaction. At its core, a standard 9x9 Sudoku grid is a system governed by a precise set of exclusionary rules. The puzzle demands that every cell \(C_{r,c}\) be assigned a value \(v \in \{1...9\}\) such that no value is repeated within any of the three fundamental "units" or "houses": the Row, the Column, and the Block.
Solving Sudoku is not merely a process of placing digits; it is a subtractive process of candidate elimination. The solver begins with a full universe of possibilities—potentially 729 candidates in an empty grid—and systematically whittles this domain down until only one valid configuration remains.
Among the hierarchy of solving strategies, the technique known as Locked Candidates (often referred to within technical communities as Intersection Removal) serves as a critical bridge. It moves the solver beyond the localized logic of a single cell or unit and forces an examination of the interactions between units. Specifically, Locked Candidates addresses the logical consequences that arise when a candidate digit is restricted to the intersection of a Block and a Line (Row or Column). This restriction creates a "lock" that acts as a forceful constraint, propagating eliminations to other parts of the grid.[1, 2]
2. Theoretical Foundations and Set Theory
To understand Locked Candidates at a professional level, one must move beyond the intuitive notion of "scanning" and engage with the underlying set theory that validates the logic. The technique is fundamentally an application of the Principle of Intersection.
2.1 The Geometry of Intersections
A standard Sudoku grid is composed of 27 distinct units: 9 Rows, 9 Columns, and 9 Blocks. Every individual cell is a member of exactly three units. Consequently, any given Block intersects with three specific Rows and three specific Columns. These intersections are not merely geometric features; they are shared subsets of cells that link the constraints of the linear units to the constraints of the box units.
2.2 The Logic of Mandatory Placement
The validity of the Locked Candidates technique rests on the Pigeonhole Principle applied to constraints. If the set of all possible locations for digit k in unit A is contained entirely within the intersection A \(\cap\) B, then the digit k must exist in the intersection. Since the digit must exist in the intersection, and the intersection is a subset of unit B, the digit k cannot exist anywhere in unit B outside of that intersection.
2.3 Taxonomy and Nomenclature
| Technical Term | Common Name | Direction of Logic | Description |
|---|---|---|---|
| Type 1 | Pointing | Block \(\rightarrow\) Line | Candidates in a Block are restricted to a single Line. Eliminations occur in the Line outside the Block. |
| Type 2 | Claiming | Line \(\rightarrow\) Block | Candidates in a Line are restricted to a single Block. Eliminations occur in the Block outside the Line. |
3. Locked Candidates Type 1: Pointing
Type 1, widely known as "Pointing," exploits the human visual tendency to process information within the compact 3x3 boxes before scanning the longer rows and columns.
3.1 Mechanism of Action
- Observation: The solver examines a specific Block.
- Constraint Identification: The solver isolates a specific digit k and identifies all cells in that Block where k is valid.
- Pattern Recognition: The solver notices that all such cells are collinear (within a single Row or Column).
- Inference: Since the Block must contain k, and the only places for k are inside the Line, the solution for k in that Line is forced to be inside the Block.
- Elimination: Consequently, k cannot appear in that Line outside of that Block.
3.2 Visual Analysis: Pointing Pairs
Consider the grid below focusing on the top-left (Block 1). We are analyzing digit 4.
2 4
4 5
4 8
Figure 1: Pointing Pair on digit 4 in Block 1 (Blue) eliminates 4s in
Block 2 (Red).
3.3 Visual Analysis: Pointing Triples
Pointing is not limited to pairs. Consider Block 5 (Center) and Column 5. We track the digit 5.
5 9
2 5
5 8
3 5
1 5
Figure 2: Pointing Triple. Digit 5 is locked in Block 5 (Blue),
eliminating vertical 5s (Red).
4. Locked Candidates Type 2: Claiming (Box-Line Reduction)
Claiming is the logical inverse of Pointing. Instead of looking for what a box tells a line, one must look for what a line tells a box.
4.3 Visual Analysis: Claiming (Row)
Let us examine Row 5 across Blocks 4, 5, and 6. The digit of interest is 7.
In this example, Row 5 contains 7s only in the middle block (Block 5). Therefore, Row 5 "claims" the 7 for Block 5, and we eliminate 7s from the rest of Block 5.
1 7
2 7
3
1
4 7
7 8
7 9
6
5 7
6
8
Figure 3: Claiming (Row). Row 5 claims the 7s, eliminating them from
the rest of Block 5 (Red).
4.4 Visual Analysis: Claiming (Column)
Logic applies vertically. Consider Column 8 and Block 6. The digit is 2.
2 3
1 2
4
5
2 5
2 6
8
2 9
7
Figure 4: Claiming (Column). Column 8 locks the 2s into Block 6
(Blue). 2s are removed from remainder of Block 6 (Red).
5. The "Fish" Taxonomy
Locked Candidates are mathematically classified as a Size 1 Fish or Cyclops Fish. This places them at the base of the complexity pyramid, below X-Wings (Size 2) and Swordfish (Size 3). Understanding that Locked Candidates are "degenerate" Fish helps advanced solvers realize that higher-order techniques are simply multi-dimensional versions of this same intersection logic.
6. Human Factors: Strategies for Detection
- Cross-Hatching: Integrate detection into the basic scanning routine. If a number doesn't result in a single placement but is restricted to a line, pause and apply Pointing logic.
- Snyder Notation: Marking candidates only when they are restricted to two cells in a block makes Pointing Pairs visually "pop" out. However, Snyder notation is less effective for Claiming (Type 2).
- Visual Filtering: Using digital apps to highlight a single number across the grid makes "lines" of candidates (Pointing) or "clusters" in a line (Claiming) much easier to spot.
7. Conclusion
The Locked Candidates technique is more than a mere trick; it is a fundamental expression of the logical constraints that define Sudoku. Whether viewed through the lens of set theory (Intersection Removal) or taxonomy (Cyclops Fish), the core principle remains the same: Restriction in one unit compels exclusion in another.